@PsubondYes and no. Mostly no though. 😉
They could have persistent servers that would idle if nobody is playing. That would waste resources and therefore money.
But they would absolutely not NEED to have those. What can be done is to have the current system with the on-demand server instances but instead of killing off the instance after the round ends they could keep the instance alive and let it progress (with all players connected) to the next map. This is EXACTLY what is being done with Portal at the moment. You don't have to join a new server after each Portal round but you can play the whole rotation. Those server instances would be killed off if the last player leaves. But in the end this would not cost any more money than the current match making system. They prove that they can do that, even without a game update by creating the Rush and Rush 2042 map rotations. They could do exactly the same with CQ / CQ large for the 2042 maps (preferable removing Hourglass in the process) if they wanted to.